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National Advisory Committee for Aeronautics, Technical Notes - The Method of Characteristics for the Determination of Supersonic Flow Over Bodies of Revolution at Small Angles of Attack

The method of characteristics has been applied for the determination
of the supersoniceflow properties around bodies of revolution at a small
angle of attack. The system developed considers the effect of the vari—
ation of entropy due to the curved shock and determines a flow that
exactly satisfies the boundary conditions in the limits of the simpli-
fications assumed. Two practical methods for numerical calculations are
given.

For the determination of aerodynamic properties of bodies of revo—
lution at supersonic speeds, two methods have been used: a method that
uses the small—disturbances theory and a method that uses the charac—
teristics theory. Both methods are successful in the determination of
the flow properties for bodies at zero angle of attack, but the precision
of the small—disturbances theory decreases when a body of revolution at
an angle of attack is considered.

For bodies of revolution having supersonic flow everywhere, the
theory of characteristics can also be used at an angle of attack.
The method of characteristics for the determination of the flow
field around bodies of revolution at an angle of attack was first used
by Ferrari (reference 1) in 1936. Ferrari considers the flow as potential
flow and develops a method for the analysis of the flow field around a
body that in the approximation of potential flow appears to be general
and can be applied to bodies of any shape and with any angle of attack.

In the determination of the flow properties along the first character—
istic Surface from.which the analysis starts, however, Ferrari analyzes
the flow around a cone of revolution; and in this part of the analysis
only small values of angle of attack are considered.

Bauer in l9h2 (reference 2) considers the same problems and shows
that for small values of angle of attack, the analysis of the flow
field around a body of revolution can be made by applying the charac—
teristics method only in one meridian plane; and, therefore, Sauer uses
characteristic lines in place of the characteristic Surfaces considered
by Ferrari. Sauer, in the development of his system, is interested
essentially in the.analysis of the flow around circular cones; and when
the method is applied to bodies of revolution of shapes different from
cones, the boundary conditions are no longer satisfied. The flow obtained
from the solution used, also at small angles of attack, vets a body that
is not a body of revolution. The body can be obtained from the body of
revolution considered initially by curving its axis of symmetry. Sauer
also assumes that the flow is potential flow. With this assumption, the
flow must be considered as potential flow for the case of the body at
zero angle of attack also; therefore, all the effects of entropy gradients
are neglected.